FORM OF TANGENTIAL EQUATION. 
431 
Again, since OP . OP'— r 2 , the points P, P' are inverse points with respect to the circle J, 
and the perpendicular through P to the line OP will be the polar of P' ; therefore the 
envelope of this perpendicular will be the reciprocal of the bicircular quartic. Now, let 
<p be the angle which the perpendicular makes with the axis of x, or the directing line, 
and v = intercept, then we have <p=10— a, and % = v sin <p. 
Therefore the tangential equation of the reciprocal of the bicircular quartic is 
v 1 sin 2 <p-\-r 1 = c L{*>J a? sin 2 £>+$ 2 cos 2 <p — (/"sin + ^ cos <p)}v sin <p. . . (271) 
Cor. The equation (271) is also the first negative pedal of a bicircular quartic. 
129. If we divide the equation (271) by sin 2 ?), and then change v into — * and cot?) 
into we get, after a slight reduction, 
\rX\*+p 2 )+v 2 -2fvX-2g\(*\ 2 =4(aW+iyy\ .... (272) 
which is in the ordinary form of tangential equations. 
130. From the equation (271) it is plain that to each value of <p there are two values 
of v. This is otherwise evident ; for erecting the perpendicular PK and P'K' to OP 
Fig. 21. 
and OP', these perpendiculars will make intercepts on the director line, rvhich will be 
the required values of v. Let C be the centre of the generating circle, then C will be a 
point on the focal conic, and CP, CP' will be normals to the quartic, and PV, P'V will 
be tangents. Now if K' be the point of contact of P'K' with its envelope, then the 
angle P'K'0 = 0P'V, and therefore OK' is parallel to CP, and OK to CP'. Hence, 
drawing from the point O two parallels to the normals at P, P', they will meet PK, P'K' 
in the points of contact of these lines with their envelopes, and they will intersect the 
tangents PY, P'V perpendicularly in the points L, L'. 
Cor. 1. The locus of the points L, L' is evidently the first positive pedal of the bicircular 
quartic. 
Cor. 2. The first positive pedal of a bicircular quartic is the inverse of its first 
negative pedal ; for evidently 
MDCCCLXXVII. 
OL' . OK'=OP . OP'=r 2 . 
3 p 
